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Communications Engineering 1
Outline
IntroductionSignal, random variable, random process and spectraAnalog modulationAnalog to digital conversionDigital transmission through bandlimited channelsSignal space representationOptimal receiversDigital modulation techniquesChannel codingSynchronizationInformation theory
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Digital transmission through bandlimitedchannels
Digital waveforms over band-limited baseband channelsBand-limited channel and Inter-symbol interferenceSignal design for band-limited channelSystem designChannel equalization Chapter 10.1-10.5
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Digital transmission through bandlimitedchannels
Band-limited channel
Modeled as a linear filter with frequency response limitedto certain frequency range
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Digital transmission through bandlimitedchannels
Baseband signaling waveforms To send the encoded digital data over a baseband
channel, we require the use of format or waveform for representing the data
System requirement on digital waveforms1. Easy to synchronize2. High spectrum utilization efficiency3. Good noise immunity4. No DC component and little low frequency
component5. Self-error-correction capability6. Et al.
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Digital transmission through bandlimitedchannels
Basic waveforms On-off or unipolar signaling Polar signaling Return-to-zero signaling Bipolar signaling (useful because no DC) Split-phase or Manchester code (no DC) Et al.
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Digital transmission through bandlimitedchannels
Basic waveforms
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Digital transmission through bandlimitedchannels
Spectra of baseband signals Consider a random binary sequence g0(t) – 0, g1(t) – 1 The pulses g0(t) and g1(t) occur independently with
probabilities given by p and 1-p, respectively. The duration of each pulse is given by Ts.
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Digital transmission through bandlimitedchannels
Spectra of baseband signals PSD of the baseband signal s(t) is[1]
1st term is the continuous freq. component2nd term is the discrete freq. component
For polar signaling with and p=1/2
For unipolar signaling with and p=1/2, and g(t) is a rectangular pulse
1. R.C. Titsworth and L. R. Welch, “Power spectra of signals modulated by random and psedurandom sequences,” JPL, CA, Technical Report, Oct. 1961.
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Digital transmission through bandlimitedchannels
Spectra of baseband signals For return-to-zero unipolar signaling
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Digital transmission through bandlimitedchannels
Inter-symbol interference The filtering effect of the band-limited channel will
cause a spreading of individual data symbols passing through
For consecutive symbols, this spreading causes part of symbol energy to overlap with neighboring symbols, causing inter-symbol interference
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Digital transmission through bandlimitedchannels
Baseband signaling through band-limited channels
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Digital transmission through bandlimitedchannels
Baseband signaling through band-limited channels
Pulse shape at the receiver filter output
Overall frequency response
Receiving filter output
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Digital transmission through bandlimitedchannels
Baseband signaling through band-limited channels
Sample the receiver filter output v(t) at tm=mT to detect Am
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Digital transmission through bandlimitedchannels
Baseband signaling through band-limited channels
Sample the receiver filter output v(t) at tm=mT to detect Am
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Digital transmission through bandlimitedchannels
Eye diagram Distorted binary wave
Eye diagram
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Digital transmission through bandlimitedchannels
ISI minimization Choose transmitter and receiver filters which shape the
received pulse function to eliminate or minimize interference between adjacent pulses, hence not to degrade the bit error rate performance of the link
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Nyquist condition for zero ISI for pulse shape p(t)
With the above condition, the receiver output simplifies to
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Nyquist’s first method for eliminating ISI is to use
The minimum transmission bandwidth for zero ISI. A channel with bandwidth B0 can support a maximal transmission rate of 2B0 symbols/sec
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Challenges of designing such p(t) or P(f)
1. P(f) is physically unrealizable due to the abrupt transitions at B0
2. p(t) decays slowly for large t, resulting in little margin of error in sampling times in the receiver
3. This demands accurate sample point timing-a major challenge in current modem/data receiver design
4. Inaccuracy in symbol timing is referred to as timing jitter.
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Raised cosine filter. P(f) is made up of 3 parts: pass band, stop band, and
transition band. The transition band is shaped like a cosine wave.
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Raised cosine filter.
The sharpness of the filter is controlled by a. Required bandwidth B=B0(1+a).
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Raised cosine filter. Taking the inverse Fourier transform
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Digital transmission through bandlimitedchannels
Signal design for band-limited channel zero ISI Raised cosine filter. Small a: higher bandwidth efficiency Large a: simpler filter with fewer stages hence easier
to implement; less sensitive to symbol timing accuracy
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Relax the condition of zero ISI and allow a controlled
amount of ISI Then we can achieve the maximal symbol rate of 2W
symbols/sec The ISI we introduce is deterministic or controlled;
hence it can be taken into account at the receiver
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal. Let {ak} be the binary sequence to be transmitted. The
pulse duration is T. Two adjacent pulses are added together, i.e., bk=ak+ak-1
The resulting sequence {bk} is called duobinary signal
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal: frequency domain.
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal: time domain.
g(t) is called a duobinary signal pulse g(0)=g0=1 g(T)=g1=1 g(iT)=gi=0,i ≠1
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal: decoding. Without noise, the received signal is the same as the
transmitted signal
When {ak} is a polar sequence with values +1 or -1
When {ak} is a unipolar sequence with values 1 or 0
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal: decoding. To recover the transmitted sequence, we can use
although the detection of the current symbol relies on the detection of the previous symbol error propagation will occur
How to solve the ambiguity problem and error propagation?
Precoding: Apply differential encoding on {ak} so that
Then the output of the duobinary signal system is
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Duobinary signal: decoding. Block diagram of precoded duobinary signal
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Modified duobinary signal
After LPF H(f), the overall response is
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Modified duobinary signal. The magnitude spectrum is a half-sin wave and hence
easy to implement No DC component and small low freq. component At sampling interval T, the sampled values are
g(t) decays as 1/t2. But time offset may cause significant problem.
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Digital transmission through bandlimitedchannels
Signal design with controlled ISI – partial response signals Modified duobinary signal: decoding. To overcome error propagation, precoding is also
needed ck=ak⊕ck-2
The coded signal is
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Digital transmission through bandlimitedchannels
Update We have discussed
1. Pulse shapes of baseband signal and their power spectrum
2. ISI in band-limited channels3. Signal design for zero ISI and controlled ISI
We will discuss system design in the presence of channel distortion1. Optimal transmitting and receiving filters2. Channel equalizer
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Digital transmission through bandlimitedchannels
Optimal transmit/receive filter Recall that when zero-ISI condition is satisfied by p(t)
with raised cosine spectrum P(f), then the sampled output of the receiver filter is Vm=Am+Nm (assume p(0)=1)
Consider binary PAM transmission: Am = ±d Variance of
with
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Digital transmission through bandlimitedchannels
Optimal transmit/receive filter Compensate the channel distortion equally between the
transmitter and receiver filters
Then, the transmit signal energy is given by
Hence
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Digital transmission through bandlimitedchannels
Optimal transmit/receive filter Noise variance at the output of the receive filter is
Special case: Hc(f)=1 for |f|≤W1. This is the idea case with “flag” fading2. No loss, same as the matched filter receiver of
AWGN channel
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Digital transmission through baseband channels
Optimal transmit/receive filter Exercise. Determine the optimum transmitting and receiving
filters for a binary communication system that transmits data at a rate R=1/T=4800 bps over a channel with a frequency response , |f|≤W where W=4800 Hz
The additive noise is zero mean white Gaussian with spectral density N0/2=10-15 Watt/Hz
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Digital transmission through baseband channels
Optimal transmit/receive filter Exercise. Since W=1/T=4800, we use a signal pulse with a
raised cosine spectrum and a roll-off factor =1. Thus,
Therefore
One can now use these filters to determine the amount of transmit energy required to achieve a specified error probability.
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Digital transmission through bandlimitedchannels
Performance with ISI If zero-ISI condition is not met, then
Let
Then
Often only 2M significant terms are considered. Hence
Finding the probability of error?
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Digital transmission through bandlimitedchannels
Performance with ISI Monte Carlo simulation.
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Digital transmission through bandlimitedchannels
Performance with ISI Monte Carlo simulation. If one want Pe to be within 10% accuracy, how many
independent simulation runs do we need? If Pe~10-9 (this is typically the case for optical
communication systems), and assume each simulation run takes 1 ms, how long will the simulation take?
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Digital transmission through bandlimitedchannels
Update Monte Carlo simulation. We have shown that by properly designing the
transmitting and receiving filters, one can guarantee zero ISI at sampling instants, thereby minimizing Pe.
Appropriate when the channel is precisely known and its characteristics do not change with time.
In practice, the channel is unknown or time-varying
We next consider channel equalizer.
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Digital transmission through bandlimitedchannels
Equalizer A receiving filter with adjustable frequency response
to minimize/eliminate inter-symbol interference
Overall frequency response
Nyquist criterion for zero-ISI
Thus, ideal zero-ISI equalizer is an inverse channel filter
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Digital transmission through bandlimitedchannels
Equalizer Linear transversal filter Finite impulse response (FIR) filter
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Digital transmission through bandlimitedchannels
Equalizer Zero-forcing (ZF) equalizer
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Digital transmission through bandlimitedchannels
Equalizer Zero-forcing (ZF) equalizer In matrix form
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Digital transmission through bandlimitedchannels
Equalizer Example.
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Digital transmission through bandlimitedchannels
Equalizer Example. By inspection
The channel response matrix
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Digital transmission through bandlimitedchannels
Equalizer Example. The inverse of this matrix
Therefore, Equalized pulse response It can be verified
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Digital transmission through bandlimitedchannels
Equalizer Example. Note that values of peq(n) for n<-2 or n>2 are not zero.
For example
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Digital transmission through bandlimitedchannels
Equalizer Minimum mean-square error equalizer. Drawback of ZF equalizer: ignores the additive noise Suppose we relax zero ISI condition, and minimize the
combined power in the residual ISI and additive noise at the output of the equalizer.
Then, we have MMSE equalizer, which is a channel equalizer optimized based on the minimum mean square error (MMSE) criterion
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Digital transmission through bandlimitedchannels
Equalizer Minimum mean-square error equalizer.
The output is sampled at t=mT:
Let Am=desired equalizer output
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Digital transmission through bandlimitedchannels
Equalizer Minimum mean-square error equalizer.
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Digital transmission through bandlimitedchannels
Equalizer MMSE equalizer vs. ZF equalizer. Both can be obtained by solving similar equations. ZF equalizer does not consider the effects of noise MMSE equalizer is designed so that mean-square error
(consisting of ISI terms and noise at the equalizer output) is minimized
Both equalizers are known as linear equalizers
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Digital transmission through bandlimitedchannels
Equalizer Decision feedback equalizer (DFE) DFE is a nonlinear equalizer which attempts to
subtract from the current symbol to be detected the ISI created by previously detected symbol
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Digital transmission through bandlimitedchannels
Equalizer Example of channels with ISI.
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Digital transmission through bandlimitedchannels
Equalizer Frequency response.
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Digital transmission through bandlimitedchannels
Equalizer Performance of MMSE equalizer.
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Digital transmission through bandlimitedchannels
Equalizer Performance of DFE equalizer.
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Digital transmission through bandlimitedchannels
Equalizer Maximum likelihood sequence estimation (MLSE).
Let the transmitting filter have a square root raised cosine frequency response
The receiving filter is matched to the transmitter filter with
The sampled output from receiving filter is
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Digital transmission through bandlimitedchannels
Equalizer Maximum likelihood sequence estimation (MLSE). Assume ISI affects finite number of symbols with
Then, the channel is equivalent to a FIR discrete-time filter
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Digital transmission through bandlimitedchannels
Equalizer Performance of MLSE
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Digital transmission through bandlimitedchannels
Equalizer
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Digital transmission through bandlimitedchannels
Suggested reading Chapter 10.1-10.5